Proof of Definite Integral

Let’s assume that f is continuous and positive on the interval [a, b] . Then the definite integral \int^b_a f(x) dx represents the area of the region bounded by the graph of f and the x-axis, from x = a to x = b. First, we partition the interval [a, b] into n subintervals, each of width \Delta x = (b - a)/n such that a = x_0 < x_1 < x_2 < . . . < x_n = b Then we can form a trapezoid for each subinterval and the area of the ith trapezoid = [\frac{f(x_{i-1}) + f(x_i)}{2}](\frac{b-a}{n}) . This implies that the sum of the areas of the n trapezoids is Area = \frac{b - a}{2n}[f(x_0) + 2f(x_1) + 2f(x_2) + ... + 2f(x_{n-1}) + f(x_n)] = \frac{b - a}{2n}[f(x_0) + f(x_n) + 2\displaystyle\sum\limits_{i=1}^{n-1} f(x_i)] = \frac{b - a}{2n}(f(x_0) + f(x_n)) + \displaystyle\sum\limits_{i=1}^{n-1} f(x_i)(\frac{b - a}{n}) = \frac{b - a}{2n}(f(x_0) + f(x_n) - 2f(x_n)) + \displaystyle\sum\limits_{i=1}^{n} f(x_i)(\frac{b - a}{n}) - 2f(x_n)(\frac{b - a}{2n}) = \frac{b - a}{2n}(f(x_0) - f(x_n)) + \displaystyle\sum\limits_{i=1}^{n} f(x_i)\Delta x = \lim_{n\to\infty}\frac{b - a}{2n}(f(x_0) - f(x_n)) + \lim_{n\to\infty}\displaystyle\sum\limits_{i=1}^{n} f(x_i)\Delta x = 0 + \lim_{n\to\infty}\displaystyle\sum\limits_{i=1}^{n}f(x_i)\Delta x = \int^b_a f(x) dx

Exercise on Relations

Let S and S‘ be the following subsets of the plane: S = \{(x,y) | y = x+1, 0<x<2\} and S'= \{(x,y) | y-x \in \mathbb{Z} \}

a) Show that S’ is an equivalence relation on the real line and that S \subset S'.

Proof: Reflexivityx-x \in \mathbb{Z}, \forall x \in \mathbb{R}

             Symmetryz \in \mathbb{Z} \Rightarrow -z \in \mathbb{Z}

             Transitivity- If x~y, y~z then z-y=(z-x)-(x-y) and thus z-y is the difference of two integers which implies that z-y is itself an integer.

To show that S \subset S' we note that y=x+1 \Rightarrow y-x=1 which \in \mathbb{Z}

b) Show that given any collection of equivalence relations on a set A, their intersection is an equivalence relation in A.

Proof: Let \{R_\alpha\}_\alpha\in A be a nonempty class of equivalence relations and let \Omega = \cap_{\alpha \in A}R_\alpha

Reflexivity- If (x,y) \in R_\alpha, \forall_\alpha \in A then (x,y) \in \Omega \Rightarrow (y,x) \in R_\alpha, \forall_\alpha \in a \Rightarrow (y,x) \in \Omega .

Symmetry(x,x) \in R_\alpha, \forall_\alpha \in A \Rightarrow (x,x) \in \Omega .

Transitivity– If (x,y), (y,z) \in R_\alpha, \forall_\alpha \in A then (x,y), (y,z) \in \Omega \Rightarrow (x,z) \in R_\alpha, \forall_{\alpha \in A} \Rightarrow (x,z) \in \Omega \therefore the intersection is an equivalence relation on A.